Part3: Other designs Prepared by: Paul Funkenbusch, Department of - PowerPoint PPT Presentation

Part3: Other designs Prepared by: Paul Funkenbusch, Department of Mechanical Engineering, University of Rochester

 Extending what you’ve learned  3+ level factors ◦ Continuous ◦ Discrete  Fractional factorials DOE mini-course, part 3, Paul Funkenbusch, 2015 2

 What if you want to test 3 or more levels of a factor?  What if a full factorial design is too large?  Build on what you’ve learned so far ( full factorial designs with 2-levels) to understand these situations.  Can get complex  only scratch surface here. ◦ Won’t go through calculations  software/textbook DOE mini-course, part 3, Paul Funkenbusch, 2015 3

Leve vel  Can code levels with ◦ -1, 0, +1 Factor -1 0 +1 ◦ 1, 2, 3, 4… 50 -- 100 X1. Temp. (  C) ◦ arbitrary (won’t use coding to determine interactions) X2. Pressure (Pa) 1 1.5 2  Construct designs by listing all combinations. TC TC X1 X1 X2 X2 y 1 -1 -1 y 1  Required # of TCs is 2 -1 0 y 2 product of number of levels for each factor. 3 -1 +1 y 3 4 +1 -1 y 4 One 2-level and one 3- 5 +1 0 y 5 level factor  2x3= 6 TC 6 +1 +1 y 6 DOE mini-course, part 3, Paul Funkenbusch, 2015 4

 # of TCs in a full factorial = product of # levels in each factor.  Effort (# of TCs) increases sharply with the number of levels per factor. Five 2-level factors Example with more levels  Five 2-level factors  One 2-level, two 3- level, and two 4-level factors  2x2x2x2x2 = 32 TC  2x3x3x4x4 = 288 TC DOE mini-course, part 3, Paul Funkenbusch, 2015 5

 Continuous factor : test non-linear effects ◦ include quadratic as well as linear terms in model ◦ 3-level is enough to do this  Discrete factor: test more than 2 possibilities ◦ human blood type (O, A, B, AB) ◦ titanium vs. steel vs. aluminum ◦ ethnically correlated differences in bone geometry ◦ Toyota vs. Ford vs. Volkswagen vs. Kia ◦ could require any number of levels  Continuous vs. discrete  affects analysis DOE mini-course, part 3, Paul Funkenbusch, 2015 6

 Can model by including quadratic terms  Example: two 3-level continuous factors ◦ 9 TC = 9 DOF = 9 potential model constants ◦ y pred = a o +a 1 X 1 +a 11 (X 1 ) 2 +a 2 X 2 +a 22 (X 2 ) 2 +a 12 X 1 X 2 +a 112 (X 1 ) 2 X 2 +a 122 X 1 (X 2 ) 2 +a 1122 (X 1 ) 2 (X 2 ) 2  m*  1 DOF (a o )  Linear factor terms  2 DOF (a 1 , a 2 )  Quadratic factor terms  2 DOF (a 11 ,a 22 )  Interaction terms  4 DOF (a 12 ,a 112 , a 122 ,a 1122 )  Often “pool” some of the (higher -order) interaction terms to estimate error ◦ e.g. a 112 , a 122 ,a 1122 DOE mini-course, part 3, Paul Funkenbusch, 2015 7

 Important to keep levels evenly spaced ◦ For example 100, 200, 300 ◦ Not 100, 150, 300  If levels are not evenly spaced, linear and quadratic terms will not be fully independent ◦ Different possible combinations of constants will fit data equally well  can’t fully distinguish linear and quadratic effects DOE mini-course, part 3, Paul Funkenbusch, 2015 8

 1 DOF for each model term ◦ Separately evaluate linear Example: two 3-level factors  9 DOF Pooled 3 higher-order interaction terms and quadratic terms ◦ Can pool some of the Source ce MS MS DOF SS SS F p (higher-order) interaction X1 16 1 16 16 0.028 terms to estimate error (X 1 ) 2 4 1 4 4 0.139 X 2 18 1 18 18 0.024 For a = 0.05 (X 2 ) 2 12 1 12 12 0.041 X1  linear term significant X 1 X 2 2 1 2 2 0.252 error 3 3 1 - - X2  linear and quadratic Total 55 8 - - - terms significant DOE mini-course, part 3, Paul Funkenbusch, 2015 9

 Can’t fit separate (linear or quadratic) terms ◦ combine factor information in a single measure ◦ # of DOF = (# of levels – 1) ◦ e.g. 3-level factor  2 DOF; 4-level factor  3 DOF  Also combine all interactions terms for factors ◦ # of DOF = product of DOF for each of the factors involved ◦ e.g. interaction between 3-level and 4-level factors  (3-1)(4-1) = 6 DOF for interaction  Example: two 3-level discrete factors (X1 and X2) ◦ 9 TC = 9 DOF  m*  1 DOF  X1  2 DOF  X2  2 DOF  X1 X2 interaction  4 DOF DOE mini-course, part 3, Paul Funkenbusch, 2015 10

 Don’t have to separate linear and quadratic terms ◦ Not concerned about building a model from the results ◦ Compare overall importance of effects (e.g. continuous factor vs. discrete factor)  In this case just treat the continuous factor as though it is discrete  Standard approach used in Taguchi methods DOE mini-course, part 3, Paul Funkenbusch, 2015 11

 All DOF for each factor and interaction are combined Example: two 3-level factors with one replication  18 DOF  Critical F may vary among sources due to differences Source ce MS DOF SS MS SS F p in # of DOF X1 20 2 10 5 0.035 ◦ Be cautious in comparing F X 2 16 2 8 4 0.057 values, judging significance X 1 X 2 32 4 8 4 0.039 For a = 0.05 error 18 9 2 - - X1  significant Total 86 17 - - - X2  not sig. (just above) Same F value, but critical F is different, X1X2  significant because # of DOF is different. DOE mini-course, part 3, Paul Funkenbusch, 2015 12

 FEM study to examine the effect that disease/age related declines in bone mechanical properties could have on femoral neck fracture risk.  Calculation  “error” = modeling error  Relative importance: cortical vs. trabecular modulus; linear vs. quadratic effects, interaction  % SS Courtesy of Mr. Alexander Kotelsky DOE mini-course, part 3, Paul Funkenbusch, 2015 13

 Strain decreases (becomes less negative) as modulus increases  Trabecular modulus has much larger effect than cortical  Noticeable curvature for trabecular modulus DOE mini-course, part 3, Paul Funkenbusch, 2015 14

 Linear fit on trabecular modulus, X2, dominates  Linear fit on cortical modulus, X1, and curvature of trabecular, (X2) 2 , are comparable  Together these three effects account for > 99% of variance DOE mini-course, part 3, Paul Funkenbusch, 2015 15

 “Fractions” of full factorials ◦ For 2- level designs: ½ or ¼ … the # of TC ◦ For 3- level designs: 1/3 or 1/9…the # of TC  Smaller experiment, but with the same number of effects ◦ Effects are “confounded” with each other ◦ Can’t separate confounded effects ◦ Use sparsity of effects  assume interaction (especially higher-order interaction) terms are zero DOE mini-course, part 3, Paul Funkenbusch, 2015 16

Full factorial Fractional factorial (eliminate ½ of TC and renumber) (there factors, 2-levels) TC TC X1 X1 X2 X2 X3 X3 y TC TC X1 X1 X2 X2 X3 X3 y 1 -1 -1 -1 y 1 1 -1 -1 +1 y 1 2 -1 -1 +1 y 2 2 -1 +1 -1 y 2 3 -1 +1 -1 y 3 3 +1 -1 -1 y 3 4 -1 +1 +1 y 4 4 +1 +1 +1 y 4 5 +1 -1 -1 y 5 6 +1 -1 +1 y 6 7 +1 +1 -1 y 7 8 +1 +1 +1 y 8 DOE mini-course, part 3, Paul Funkenbusch, 2015 17

Fractional factorial Confounding  In this design X3  X1X2 ◦ Therefore: D x3  D x1x2 (calculation) ◦ X3 and X1X2 are confounded ◦ Can’t distinguish which causes the measured effect ◦ Assume X1X2 interaction = 0 ◦ Determine D x3  4 responses (4 y’s)  8 “effects”  Similarly m* , D x1 , D x2 , D x3 , D x1x2 , D x1x3 ,  ◦ X1  X2X3  confounded D x2x3 , D x1x2x3 ◦ X2  X1X3  confounded  Assume interactions are zero ◦ X1X2X3  +1  confounded with m* to determine other effects DOE mini-course, part 3, Paul Funkenbusch, 2015 18

 Must preserve symmetry of design  Can produce different confounding patterns ◦ i.e. what confounds with what  “Resolution” is one measure of the severity of confounding ◦ higher values indicate less severe confounding ◦ III is lowest (worst) level ◦ In practice III, IV, and V resolutions are common  Described in most introductory textbooks  Most software packages will also produce designs for you DOE mini-course, part 3, Paul Funkenbusch, 2015 19

 Upfront assumption that some effects (generally higher-order interactions) are zero  Other wise the same as for full factorials DOE mini-course, part 3, Paul Funkenbusch, 2015 20

 Nine 2-level factors Level el Factor -1 +1  Full-factorial X1. # of cooling ports 1 3 ◦ 2 9 = 512 TC X2. Diamond grit fine coarse ◦ Too large! X3. Bur diameter (mm) 0.14 0.18 X4. Bur type multi-use one-use  Fractional factorial X5. Length of cut (mm) 2 9 ◦ 2 9-4 = 32 TC X6. Cut type central tang. X7. RPM 400k 300k ◦ 1/16 of full factorial X8. Target load (gf) 75 150  Resolution IV design X9. Coolant rate (ml/min) 10 50  Analysis assumed all Based on work by Mario Rotella et al. interactions are zero DOE mini-course, part 3, Paul Funkenbusch, 2015 21

ANOM     m average response at level 1  1     m average response at level 1  1 ANOVA n      SS = D 2 *(# of TC)/4  2 Total SS = m * y i i = 1 error term by subtraction DOE mini-course, part 3, Paul Funkenbusch, 2015 22